Editing SSC/VirialStability (section)

=Virial Stability of BiPolytropes=

==Discussion with Kundan Kadam==
See particularly &hellip; [https://ui.adsabs.harvard.edu/abs/2016MNRAS.462.2237K/abstract K. Kadam ''et al.'' (2016, MNRAS, 462, pp. 2237-2245)]

The comments provided inside the following box are for Kundan Kadam, providing feedback on a derivation he sent to me ( J. E. Tohline) on Monday, 27 January 2014.

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[<font color="red">31 January 2014</font>] I agree with the following derived expressions; they all already appear in my presentation elsewhere on this page:
<div align="center">
Throughout the envelope &hellip; &nbsp; &nbsp; &nbsp;
<math>
M_r = \frac{4\pi}{3} \biggl( r^3\rho_e + r_i^3\rho_c - r_i^3\rho_e\biggr) \, ;
</math>

<math>
M_\mathrm{tot} = \frac{4\pi}{3}\rho_c \biggl( \frac{q^3 R^3}{\nu} \biggr) \, ;
</math>

<math>
\nu = \biggl[1 + \biggl(\frac{\rho_e}{\rho_c}\biggr)(q^{-3} - 1) \biggr]^{-1} \, .
</math>
</div>
During the recent December holiday break, I also used the hydrostatic balance relation to derive expressions for the pressure throughout the two-component (uniform density core + uniform density envelope) model, as you have done.  I agree completely with your derivation of the expressions for &#8212; and relationships between &#8212; <math>P_{cc}</math>, <math>P_{ci}</math>, and <math>P_{ei}</math>.  In addition, I chose to normalize all the pressures to,
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<math>
P_\mathrm{norm} \equiv \biggl(G^3 \rho_c^4 M_\mathrm{tot}^2 \biggr)^{1/3} \, .
</math>
</div>
Letting an asterisk superscript denote normalized pressures, that is, <math>P^* \equiv P/P_\mathrm{norm}</math>, my derived expressions are:
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<math>
P^*_{cc} = P^*_{ci} + \biggr[\frac{\pi \nu^2}{6} \biggr]^{1/3} \, ,
</math>
</div>
and,
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<math>
P^*_{ei} = P^*_{ci} = \biggr[\frac{\pi \nu^2}{6} \biggr]^{1/3} \biggl( \frac{\rho_e}{\rho_c}\biggr) \biggl[ \frac{\rho_e}{\rho_c} 
\biggl(\frac{1}{q^2}-1\biggr) + 2 \biggl(1- \frac{\rho_e}{\rho_c}\biggr)(1-q) \biggr] \, .
</math>
</div>
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[<font color="red">4 February 2014</font>] In deriving the above analytic expression for the pressure, we have effectively determined the detailed [[SSC/Structure/BiPolytropes/Analytic00#BiPolytrope_with_nc_.3D_0_and_ne_.3D_0|equilibrium structure of a bipolytrope that has <math>n_c = 0</math> and <math>n_e = 0</math>]] without ever using the virial theorem.  This is an excellent accomplishment for a variety of reasons.  

<ins>First</ins>, we can now compare and contrast these newly derived expressions with the analogous expressions that have been derived earlier for a [[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1|bipolytrope with  <math>n_c = 5</math> and <math>n_e = 1</math>]].  For example, at the end of [[SSC/Structure/BiPolytropes/Analytic00#Step_7:__Surface_Boundary_Condition|Step #7 in the new derivation]], we find the following expressions for the equilibrium radius and total mass of a bipolytrope with <math>(n_c, n_e) = (0, 0)</math> expressed in terms of the chosen central density, <math>\rho_0</math>, and central pressure, <math>P_0</math>, of the configuration:
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<tr>
  <td align="right">
<math>
\biggl[ \frac{G\rho_0^2}{P_0} \biggr]^{1/2} R</math>
  </td>
  <td align="center">
&nbsp; <math>~=</math>&nbsp; 
  </td>
  <td align="left">
<math>\biggl( \frac{1}{2\pi} \biggr)^{1/2} \frac{\sqrt{3}}{q g} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\biggl[ \frac{G^3\rho_0^4}{P_0^3} \biggr]^{1/2} M_\mathrm{tot}</math>
  </td>
  <td align="center">
&nbsp; <math>~=</math>&nbsp; 
  </td>
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<math>\biggl( \frac{2}{\pi} \biggr)^{1/2} \frac{\sqrt{3}}{\nu g^3} \, ,</math>
  </td>
</tr>
</table>
</div>
where the function (<font color="red">note that, for a few days, there was a sign error in this definition of <math>g</math></font>),
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<math>
g(q, \rho_e/\rho_0) \equiv \biggl\{ 1  + 
\biggl(\frac{\rho_e}{\rho_0}\biggr)  \biggl[ 2 \biggl(1 - \frac{\rho_e}{\rho_0} \biggr) \biggl(1- q \biggr) + \frac{\rho_e}{\rho_0} \biggl(\frac{1}{q^2} - 1\biggr) \biggr] \biggr\}^{1/2}\, .
</math>
</div>
We will also find it useful to combine these two expressions to eliminate direct reference to the central density, <math>\rho_0</math>, obtaining,
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<tr>
  <td align="right">
<math>
\biggl[ \frac{R^4}{GM_\mathrm{tot}^2} \biggr] P_0</math>
  </td>
  <td align="center">
&nbsp; <math>~=</math>&nbsp; 
  </td>
  <td align="left">
<math>\biggl( \frac{1}{2^3\pi} \biggr) \frac{3\nu^2 g^2}{q^4} \, .</math>
  </td>
</tr>
</table>
</div>

For comparison, referring back to the "Normalization" discussion and table of "Parameter Values" provided in our [[SSC/Structure/BiPolytropes/Analytic51#Examples|example bipolytrope with <math>(n_c, n_e) = (5, 1)</math>]], we have,
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  <td align="right">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr) \biggl[ \frac{G\rho_0^2}{(K_c \rho_0^{6/5})} \biggr]^{1/2} R</math>
  </td>
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&nbsp; <math>~=</math>&nbsp; 
  </td>
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<math>\biggl( \frac{1}{2\pi} \biggr)^{1/2} \frac{\eta_s}{\theta_i^2} \, ,</math>
  </td>
</tr>

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  <td align="right">
<math>
\biggl( \frac{\mu_e}{\mu_c} \biggr) ^2 \biggl[ \frac{G^3\rho_0^4}{(K_c \rho_0^{6/5})^3} \biggr]^{1/2} M_\mathrm{tot}</math>
  </td>
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&nbsp; <math>~=</math>&nbsp; 
  </td>
  <td align="left">
<math>\biggl( \frac{2}{\pi} \biggr)^{1/2} \frac{A\eta_s}{\theta_i} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\biggl[ \frac{R^4}{GM^2_\mathrm{tot}} \biggr]^{1/2} K_c \rho_0^{6/5}</math>
  </td>
  <td align="center">
&nbsp; <math>~=</math>&nbsp; 
  </td>
  <td align="left">
<math>\biggl( \frac{1}{2^3\pi} \biggr) \frac{\eta_s^2}{A^2\theta_i^6} \, ,</math>
  </td>
</tr>
</table>
</div>
where expressions for the various functions, <math>A, \eta_s, \theta_i</math>, are also provided in the table of "Parameter Values."  The dimensional normalizations are clearly the same in both types of bipolytropes because, in the latter case, <math>P_0 = K_c \rho_0^{6/5}</math>.

<ins>Second</ins>, a comparison between these two types of bipolytropes may help clarify how a discontinuous jump in the mean molecular weight should be handled in the more general virial analysis.  Usually <math>\rho_e/\rho_c</math> is set equal to <math>\mu_e/\mu_c</math> at the interface, but if the core and envelope are both treated as having uniform densities, setting the ratio of densities to the ratio of mean molecular weights seems to overconstrain the problem.  I haven't figured out yet how to handle this, but maybe this model comparison will help.

<ins>Third</ins>, when conducting the virial analysis, it should now be clear how to evaluate the system's free energy.  When the system is treated as being composed of a central uniform-density spherical core surrounded by a uniform-density envelope, we have already derived an analytic expression for the total gravitational potential energy, <math>W</math>.  The above derivation of the pressure distribution throughout such a configuration &#8212; that is, throughout a bipolytrope with <math>(n_c, n_e) = (0, 0)</math> &#8212; now lets us derive an analytic expression for the thermal energy of the core, <math>S_\mathrm{core}</math>, and an analytic expression for the thermal energy of the envelope, <math>S_\mathrm{env}</math>.  Whether focusing on the core or the envelope, start with an appreciation that,
<div align="center">
<math>
dS = \frac{3}{2} \biggl( \frac{P}{\rho} \biggr) dm = \frac{3}{2} \biggl( \frac{P}{\rho} \biggr) \biggl( \frac{4\pi}{3} \biggr) \rho r^2 dr
= 2\pi Pr^2 dr \, .
</math>
</div>
Hence,
<div align="center">
<math>
S_\mathrm{core} = 2\pi \int_0^{r_i} [P(r)]_\mathrm{core} r^2 dr; ~~~~~\mathrm{and}~~~~~
S_\mathrm{env} = 2\pi \int_{r_i}^R [P(r)]_\mathrm{env} r^2 dr \, .
</math>
</div>
Then, for either volume segment, the relationship between the internal energy, <math>U</math>, and the thermal energy, <math>S</math>, is,
<div align="center">
<math>
U = \biggl[ \frac{2}{3(\gamma - 1)} \biggr] S \, .
</math>
</div>

My initial derivation gives,
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<font color="darkblue">
Do not use these expressions as they are flawed.  The correct expressions follow in the comments dated 12 February 2014.
</font>
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<tr><td>
<math>
S_\mathrm{core} = 2\pi P_0 R^3 \biggl[q^3 - q^5 \biggl( \frac{2\pi G\rho_0^2 R^2}{5P_0} \biggr) \biggr] \, ;
</math>
</td></tr>
<tr><td>
<math>
S_\mathrm{env} = \frac{4\pi}{3} R^3 \biggl\{ \frac{3P_i}{2} (1-q^3) + \pi G \rho_0^2 R^2 \biggl(\frac{\rho_e}{\rho_c} \biggr) 
\biggl[  \frac{1}{6}(6q^3 - q^2 - 5q^5) - \frac{1}{10}\biggl(\frac{\rho_e}{\rho_c} \biggr) (2 + 10q^3 - 5q^2-7q^5)      \biggr] \biggr\} \, .
</math>
</td></tr>
</table>

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<br />
[<font color="red">12 February 2014</font>] In an [[SSC/Structure/BiPolytropes/Analytic00#Thermal_Energy_Content|accompanying discussion of the thermal energy content of an <math>(n_c, n_e) = (0, 0)</math> bipolytrope]], I've re-derived and cross-checked the expressions for <math>S_\mathrm{core}</math> and <math>S_\mathrm{env}</math>.  I am quite confident that the correct expressions are:

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<math>~S_\mathrm{core}</math>
  </td>
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<math>~=~</math>
  </td>
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<math>\biggl( \frac{4\pi}{5} \biggr) R^3 q^5 \biggl (\frac{5P_i}{2q^2} + \Pi \biggr)  \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~S_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math> 
\frac{\pi R^3}{5} \biggl\{ 10 P_i  (1-q^3)  
+ 10 \Pi \biggl( \frac{\rho_e}{\rho_0} \biggr) \biggl[-2q^2 + 3q^3 - q^5  \biggr]
  + 6 \Pi \biggl( \frac{\rho_e}{\rho_0} \biggr)^2 \biggl[ -1 + 5q^2 -5q^3 +  q^5   \biggr] \biggr\} \, ,
</math>
  </td>
</tr>
</table>
</div>

where, <math>P_i</math> is the pressure at the interface,
<div align="center">
<math>
\Pi \equiv \frac{3}{2^3 \pi} 
\biggl( \frac{GM_\mathrm{tot}^2}{R^4} \biggr) \biggl( \frac{\nu^2}{q^6} \biggr) \, ,
</math>
</div>
and the relationship between the central pressure and the pressure at the interface is,
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<math>~P_0 = P_i + \Pi q^2 \, .</math>
</div>

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[<font color="red">7 February 2014</font>] More succinctly, this <math>(n_c, n_e) = (0,0)</math> bipolytrope is very interesting because it provides three independent ways of deriving the proper equilibrium structure.  If we work through the detailed derivation in all three cases, we should have a very firm foundation from which to perform the stability analysis of a broad range of bipolytropic configurations. The three derivations are &hellip;
<ol>
<li>Detailed force balance (Kundan has already done this):  Use the hydrostatic balance equation to determine how the pressure varies with radius throughout the two-component (core-envelope) structure.  For a given choice of <math>P_0, \rho_0, \nu,</math> and <math>q</math>, the radius and, hence, total mass of the configuration is determined by the surface boundary condition, that is, <math>P = 0</math> at <math>r = R</math>.  These results are summarized above.
<li>Virial equilibrium:  The structures derived from demanding detailed force balance should also be in virial equilibrium.  (The virial equilibrium condition is less demanding than detailed force balance because it is a global statement of energy balance and is not concerned with internal structural details. Virial equilibrium is a necessary but not sufficient condition; the force-balance solution gives the more detailed information but must, itself, also define a structure that has virial energy balance.)  It is well known that, for the simplest spherically symmetric stars, the statement of virial balance is,
<div align="center">
<math>
2S + W = 0 \, .
</math>
</div>
Since we have analytic expressions for both <math>S</math> and <math>W</math>, we should check to see if they sum appropriately to zero as expected from this simple statement of virial equilibrium.  [Note inserted <font color="red">12 February 2014</font>:  An [[SSC/Structure/BiPolytropes/Analytic00#Virial_Equilibrium|accompanying discussion of the virial equilibrium of this particular bipolytrope]] shows that the relation does hold precisely.]
</ol>
<ol start="3">
<li>Extrema in the Free Energy:  We should also expect to be able to define the free energy of this bipolytrope by the expression,
<div align="center">
<math>
\mathfrak{G} = U + W \, .
</math>
</div>
Then, for a given choice of <math>P_0, \rho_0, \nu,</math> and <math>q</math> (perhaps we should choose <math>M_\mathrm{tot}</math> instead of <math>P_0</math>), the equilibrium radius should be defined by the radius at which,
<div align="center">
<math>
\frac{\partial \mathfrak{G}}{\partial R} = 0 \, .
</math>
</div>
We should check to make sure that this is also true.
</ol>
Items #2 and #3 are the next things I will be working on.

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<br />
[<font color="red">17 February 2014; by J. E. Tohline</font>] Using derivatives of the free energy function, <math>~\mathfrak{G}</math>, I have finished deriving the equilibrium radius for bipolytropic configurations where the density of the core <math>~(\rho_0)</math> and the density of the envelope <math>~(\rho_e)</math> are both assumed to be uniform.  I have also finished deriving the condition for the dynamical stability of such configurations assuming that, as the equilibrium structure undergoes a radial perturbation, the core compresses along a <math>~\gamma_c</math> adiabat while the envelope compresses along a <math>~\gamma_e</math> adiabat.  The [[SSC/Structure/BiPolytropes/Analytic00#Equilibrium_Condition|equilibrium radius is derived here]] and the [[SSC/Structure/BiPolytropes/Analytic00#Stability_Condition|stability condition is described here]].

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<br />
[<font color="red">25 February 2014; by J. E. Tohline</font>] I have extended the free-energy-based stability analysis to models of <math>~(n_c, n_e) = (5, 1)</math> bipolytropes.  As is [[SSC/Structure/BiPolytropes/Analytic51#Expression_for_Free_Energy|detailed here]], I have been able to derive analytic, closed form expressions for the gravitational potential energy and internal thermal energies of these configurations.  As is [[SSC/Structure/BiPolytropes/Analytic51#Equilibrium_Condition|shown here]], an examination of the first derivative of the free energy identifies equilibrium configurations that exactly match those derived from the earlier detailed force-balance derivation; and, finally, as [[SSC/Structure/BiPolytropes/Analytic51#Stability_Condition|shown here (see especially Figure 3)]], the second derivative of the free energy expression allows us to determine which equilibrium models are stable and which are dynamically unstable.  Hooray!
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